Question

Two tangents PA and PB are drawn to a circle with O from an external point p. Prove that angle APB equal to 2 angle OAB

Answer 1

Answer:

Step-by-step explanation:

Join OB

We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

angle OAP = angle OBP = 90�

Now,

angle OAP + angle APB + angle OBP + angle AOB = 360� [Angle sum property of quadrilaterals]

implies 90�+ angle APB + 90� + angle AOB = 360�

implies angle AOB = 360� - 180� - angle APB = 180� - angle APB ....(1)

Now, in triangle OAB, OA is equal to OB as both are radii.

implies angle OAB = angle OBA [In a triangle, angles opposite to equal sides are equal]

Now, on applying angle sum property of triangles in ?AOB, we obtain

Angle OAB + angle OBA + angle AOB = 180�

implies 2angle OAB + angle AOB = 180�

implies 2 angle OAB + (180� - angle APB) = 180� [Using (1)]

implies 2 angle OAB = angle APB

Thus, the given result is proved